`f(x) = (((2x + 3)^2)(x - 2)^5)/((x^3)(x - 5)^2)` Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.
Vertical asymptotes are the undefined points, also known as zeros of denominator.
Let's find the zeros of denominator of the function,
Vertical Asymptotes are x=0 , x=5
For Horizontal Asymptotes
Degree of Numerator of the function=7
Degree of Denominator of the function=5
Degree of Numerator`>` 1+Degree of Denominator
`:.` There is no Horizontal Asymptote
Now let's find intercepts
x intercepts can be found when f(x)=0
`2x+3=0 , x-2=0`
`x=-3/2 , x=2`
So x intercepts are -1.5 and 2.
Since x is undefined at x=0 , there are no y intercepts.
See the attached graph and link.
From the graph,
f(8) `~~` 600``
No Local Maximum
Minimum at `(-1.5,0)`